Abstract
In this paper, the bifurcation structure and stability of the nonconstant positive steady states in a ratio-dependent predator-prey model with prey-taxis are described. First, the stability results for the positive constant equilibrium will extend to the derivative of prey's functional response with prey is positive. It is found that attractive prey-taxis can stabilize equilibrium even self-diffusion-driven instability has occurred, and repulsive prey-taxis can destabilize equilibrium. Then, several numerical simulations are performed to visualize the complex dynamic behavior: The model exhibits taxis-controlled spatial patterns such as spots pattern, spots-stripe pattern and stripe pattern. Finally, the stationary problem in one-dimensional domain is investigated. Choosing prey-taxis coefficient as bifurcation parameter, we derive the existence and stability of nonconstant positive steady states by applying Crandall–Rabinowitz bifurcation theory, and obtain the stable bifurcating solutions near the bifurcation point under suitable conditions.